A quasilinearization algorithm is proposed to solve optimal control problems with a scalar control variable and a scalar state inequality constraint using a transformation technique. The optimal control problem is transformed into an unconstrained problem with an auxiliary control and higher dimension. Owing to the appearance of singular arcs in the transformed problem, this is made non-singular by adding a quadratic function of the auxiliary control to its performance index. A proposed quasilinearization algorithm is then used to generate a series of solutions of the modified transformed problems which converges to the solution of the original state-constrained problem in the limit. A distinct feature of the proposed algorithm is that each modified transformed problem yields a feasible trajectory of the original state-constrained problem. Two examples have been solved to illustrate the effectiveness of the proposed quasilinearization algorithm.